nLab bifibration


This page is about Grothendieck fibrations that are also opfibrations. Not to be confused with two-sided fibrations nor with fibrations of 2-categories (both of which some authors also refer to as “bifibrations”).



A bifibration of categories is a functor

E B \array{ E \\ \big\downarrow \\ B }

that is both a Grothendieck fibration as well as an opfibration.

Therefore every morphism f:b 1b 2f \colon b_1 \to b_2 in a bifibration has both a push-forward f *:E b 1E b 2f_* : E_{b_1} \to E_{b_2} as well as a pullback f *:E b 2E b 1f^* : E_{b_2} \to E_{b_1}.



Relation to pseudofunctors in adjoints


Under the Grothendieck construction, the Grothendieck fibrations which arise from pseudofunctors Cat\mathcal{B} \longrightarrow Cat that factor through Cat Adj Cat_{Adj} are equivalently the bifibrations.


A Grothendieck construction on a pseudofunctor yielding a bifibration is fairly immediately equivalent to all base change functors having an adjoint on the respective side (e.g. Jacobs (1998), Lem. 9.1.2). By the fact that Cat adjCatCat_{adj} \to Cat is a locally full sub-2-category (this Prop.) this already means that the given pseudofunctor factors through Cat adjCat_adj, and essentially uniquely so.

See also Harpaz & Prasma (2015), Prop. 2.2.1.


Further factoring through ModCat Cat Ajd \longrightarrow Cat_{Ajd} hence yields bifibrations of model categories [Harpaz & Prasma (2015), Sec. 3; Cagne & Melliès (2020)]. See at model structures on Grothendieck constructions for more on this.

Relation to monadic descent

Ordinary Grothendieck fibrations correspond to pseudofunctors to Cat, by the Grothendieck construction and hence to prestacks. For these one may consider descent.

If the fibration is a bifibration, there is a particularly elegant algebraic way to encode its descent properties; this is monadic descent. The Benabou–Roubaud theorem characterizes descent properties for bifibrations.

Bifibration of bicategories

The basic theory of 2-fibrations, fibrations of bicategories, is developed in (Buckley). One (unpublished) proposal by Buckley and Shulman, for a “2-bifibration”, a bifibration of bicategories, is a functor of bicategories that is both a 2-fibration and a 2-opfibration. This should correspond to a pseudofunctor into 2Cat adj2 Cat_{adj}, just as for 1-categories. There is a difference from this case, however, in that although a 2-fibration (corresponding to a “doubly contravariant” pseudofunctor B coop2CatB^{coop} \to 2Cat) has both cartesian lifts of 2-cells and 1-cells, a 2-opfibration (corresponding to a totally covariant pseudofunctor B2CatB\to 2Cat) has opcartesian lifts of 1-cells but still cartesian lifts of 2-cells.


Original notion and terminology of “bifibration”:

  • Alexander Grothendieck, Catégories co-fibrées, catégories bi-fibrées., Section 10 in exposé VI of: Revêtements Etales et Groupe Fondamental - Séminaire de Géometrie Algébrique du Bois Marie 1960/61 (SGA 1), LNM 224 Springer (1971) [updated version with comments by M. Raynaud: arxiv.0206203]

Further early discussion (not using the terminology “bifibration”, though):

Discussion in the context of the Beck-Chevalley condition:

In the context of categorical semantics for dependent types:

Relation to pseudofunctors with values in Cat Adj Cat_{Adj} and ModCat ModCat (cf. model structures on Grothendieck constructions):

Last revised on May 6, 2023 at 16:58:21. See the history of this page for a list of all contributions to it.